Question

Americium-241 is widely used in smoke detectors. The radiation released by this element ionizes particles that are then detected by a charged-particle collector. The half-life of \(^{24} \mathrm{Am}\) is 433 years, and it decays by emitting \(\alpha\) particles. How many \(\alpha\) particles are emitted each second by a 5.00 -g sample of \(^{241} \mathrm{Am}\) ?

Short answer

The number of α particles emitted per second by a 5.00 g sample of \(^{241}Am\) can be determined as follows: 1. Calculate the number of \(^{241}Am\) atoms in the sample: Number of atoms = \( \frac{5.00 \: g}{241 \: g/mol} \) × (6.022 × 10^23 atoms/mol) 2. Determine the decay constant, λ: λ (decays/s) = \( \frac{0.693}{433 \: years} \) × \( \frac{1 \: year}{3.15 \times 10^7 \: s}\) 3. Calculate the decay rate (decays/s): Decay rate (decays/s) = λ × Number of americium-241 atoms 4. Determine the number of α particles emitted per second: Number of α particles emitted each second = Decay rate (decays/s)

Step-by-step solution

1. Determine the number of americium-241 atoms in the sample

First, we need to calculate the number of moles of americium-241 in the given 5.00 g sample. To do this, we will use the molar mass of americium-241, which is 241 g/mol. Moles of \(^{241}Am\) = \( \frac{mass}{molar \: mass} \) = \( \frac{5.00 \: g}{241 \: g/mol} \) Now, we will use Avogadro's number ( 6.022 x 10^23 atoms/mol) to convert moles into the number of americium-241 atoms: Number of atoms = Moles × Avogadro's number = \( \frac{5.00 \: g}{241 \: g/mol} \) × ( 6.022 x 10^23 atoms/mol)

2. Calculate the decay constant

Next, we need to find the decay constant, λ (lambda), using the half-life formula: Half-life (t1/2) = \( \frac{0.693}{λ} \) Solve for λ: λ = \( \frac{0.693}{t_{1/2}} \) = \( \frac{0.693}{433 \: years} \) Now, we need to convert λ to decays per second: λ (decays/s) = \( λ \frac{1 \: year}{3.15 \times 10^7 \: s}\)

3. Determine the number of decays per second by americium-241

Now that we have the decay constant, we can use it to calculate the decay rate of the sample. The decay rate is the product of the decay constant and the number of \(^{241}Am\) atoms. Decay rate (decays/s) = λ × Number of americium-241 atoms

4. Calculate the number of α particles emitted per second

Since americium-241 decays by emitting α particles, the number of α particles emitted per second will be equal to the decay rate per second found in step 3. Number of α particles emitted each second = Decay rate (decays/s)

Key concepts

Americium-241

Americium-241 is a man-made radioactive element found in the actinide series of the periodic table. It is most commonly used in household smoke detectors due to its ability to emit alpha particles, which help in ionizing the air and detecting smoke particles.
This element is created by bombarding plutonium with neutrons in a nuclear reactor. It has a long half-life of 433 years, making it relatively stable for an isotope.
Applications:

• Smoke Detectors: Americium-241 ionizes the air in the detector chamber. This allows the detector to pick up tiny particles of smoke and trigger the alarm before a fire develops fully.
• Research and Industry: Apart from smoke detectors, it's used in research for radiation and radiography purposes, offering a constant source of ionization.

Half-life Calculation

The half-life of a radioactive substance is the time it takes for half of the substance to decay. This period is key in understanding how long a radioactive isotope will remain active, affecting its application in different fields.
For Americium-241, the half-life is 433 years. Knowing the half-life is fundamental for calculating how many radioactive decays occur in a given period.
To calculate the decay constant \( \lambda \, \), we use the formula:\[ \lambda = \frac{0.693}{t_{1/2}} \] where \( t_{1/2} = 433 ext{ years} \, \). Converting years to seconds using the conversion factor, where 1 year equals approximately \( 3.15\times10^7 ext{ seconds} \, \), allows converting this value into more convenient time intervals.

Alpha Particles

Alpha particles are one type of radiation, consisting of 2 protons and 2 neutrons. They are emitted from the nucleus of a decaying atom to form a different element. This change is significant in nuclear chemistry and physics.
Alpha particles are relatively heavy, giving them the ability to interact strongly with matter, though they can be stopped by something as minimal as a sheet of paper or even human skin.
Key Characteristics of Alpha Particles:

• High Ionization: They possess a high ability to ionize atoms and molecules along their path.
• Low Penetration Power: Though they cannot penetrate deeply into materials, their ionization strength makes them hazardous if ingested or inhaled.
• Applications: Due to these properties, they have been harnessed in devices like smoke detectors and also in radiation therapy for cancer treatment.